Monte Carlo simulations and option pricing

1 Monte Carlo simulations and optionpricingby Bingqian LuUndergraduate Mathematics DepartmentPennsylvania State UniversityUniversity Park, PA 16802 Project Supervisor: Professor Anna MazzucatoJuly, 2011 AbstractMonte Carlo simulation is a legitimate and widely used technique for dealingwith uncertainty in many aspects of business operations. The purpose ofthis report is to explore the application of this technique to the stock volalityand to test its accuracy by comparing the result computed byMonte CarloEstimatewith the result ofBlack-Schole modeland the Variance ReductionbyAntitheric Variattes. The mathematical computer softwear applicationthat we use to compute and test the relationship between the sample sizeand the accuracy of Monte Carlo simulation is itshapeMathematica. It alsoprovides numerical and geometrical evidence for our Introduction to Monte Carlo SimulaionMonte Carlo option Priceis a method often used in Mathematical fi-nance to calculate the value of an option with multiple sources of uncertain-ties and random features, such as changing interest rates, stock prices orexchange rates, This method is called Monte Carlo simulation , namingafter the city of Monte Carlo , which is noted for its casinos.

2 In my project, IuseMathematica, a mathematics computer software, we can easily createa sequence of random number indicating the uncertainties that we mighthave for the stock prices for pricing Financial Options by Flipping a CoinA distcrete model for change in price of a stock over a time interval [0,T] isSn+1=Sn+ Sn t+ Sn n+1 t, S0=s(1)whereSn=Stnis the stock price at timetn=n t,n= 0,1,..,N 1, t=T/N, is the annual growth rate of the stock, and is a measure of thestocks annual price volatility or tendency to fluctuate. Highly volatile stockshave large values of . Each term in sequence 1, takes on the valueof 1 or -1 depending on the outcoming value of a coin tossing experiment,heads or tails respectively. In other words, for each n=1,2,.. n={1 with probability = 1/2 1 with probability = 1/2(2)By usingMathematica, it is very easy to create a sequence of random num-ber. With this sequence, the equation (1) can then be used to simulatea sample pathortrajectoryof stock prices,{s,S1,S2.}}

3 ,SN}. For ourpurpose here, it has been shown as a relatively accurate method of pricingoptions and very useful for options that depend on Proof of highly volatile stocks have large val-ues for = us simulate several sample trajectories of (1) for the following parametervalues and plot the trajectoris: = , = ,T= 1,s= $40,N= following figures are the graphs that we got for = Since wehave to be the value generated by flipping a coin, it gives us arbitrary valuesand thus, we have different graphs for parameter = , = ,T=1,s= $40,N= 254 This is another possibile 1: Let M be the value ofS254of different trajectories, k is the numberof = we repeated the experiment using the value of = for the volalityand other parameters remain the , it should have a number of different graphs due to the arbitraryvalue of we generated by Mathematica. The following figures are thegraphs that we got for = : From the two experiments above with the large different and constant other parameters, we can tell that the larger the , thegreater degree of variability in their behavior forthe s it is permissibleto use random number generator that creates normally distributed randomnumbers with mean zero and variance one.

4 Recall that the standard normaldistribution has the bell-shape with a standard deviation of and standardnormal random variable has a mean of Monte Carlo Method vs. Black-Scholes Monte Carlo Method and its computingMonte Carlo MethodIn the formular (1), the random termsSn n+1 ton the right-hand sidecan be consider as shocks or distrubances that model functuations in thestock price. After repeatedly simulating stock price trajectories, as we didin the previous chapter, and computing appropriate averages, it is possibleto obtain estimates of the price of aEuropean call option , a type 2:S254= , = derivative. A statistical simulation algorithm of this type is whatwe known as Monte Carlo method A European call option is a contract between two parties, a holder and awriter, whereby, for a premium paid to the writer, the holder can purchasethe stock at a future date T (the expiration date) at a price K (the strikeprice) agreed upon in the contract.

5 If the buyer elect to exercise the optionon the expiration date, the writer is obligated to sell the inderlying stockto the buyer at the price K, the strike price. Thus, the option has a payofffunctionf(S) =max(S K,0)(3)whereS=S(T) is the price of the underlying stock at the time T whenthe option expires. This equation (3) produces one possible option value atexpiration and after computing this thousands of times in order to obtain afeel for the possible error in estimating the price. Equation(3) is also knownas the value of the option at time T since ifS(T)> K, the holder can pur-chase, at price K, stock with market value S(T) and thereby make a profitequal toS(T) Knot counting the option premium. However, on the otherhand, ifS(T)< K, the holder will simply let the option expire since therewould be no reason to purchase stock at a price that exceeds the other words, the option valuation problem is determine the correctand fair price of the option at the time that the holder and writer enterinto the contract.

6 In order to estimate the price call of a call option using aMonte Carlo method, an ensemble{S(k)N=S(k)(T),k= 1,..M}(4) 3:S254= , = M stock orices at expiration is generated using the difference equationS(k)n+1=S(k)n+rS(k)n t+ S(k)n (k)n+1 t, S(k)0=s(5)Equation (5) is identical to equation (1) for eachk= 1,..,M, except thegrowth rate is replaces by the annual interest r that it costs the writerto borrow money. option pricing theory requires that the average value ofthe payoffs{f(S(k)N0,k= 1,..,M}be equal to the compounded total returnobtained by investing the option premium, C(s), at rate r over the life ofoption,1MM k=1f(s(k)N) = (1 +r t)N C(s).(6)Solving(6) for C(s) yields the Monte Carlo estimate C(s) = (1 +r t) N{1MM k=1f(s(k)N)}(7)for the option price. So, the Monte Carlo estimate C(s) is the present valueof the average of the payoffs computed using rules of compound Computing Monte Carlo EstimateWe use equation (7) to compute a Monte Carlo estimate of the value of a fivemonth call option , in other wordT=512years, for the following parametervalues:r= , = ,N= 254,andK= $50.)

7 N is the number of timesof steps for each 4:S254= , = Comparing to the Exact Black-Scholes Formu-larMonte Carlo has been used to price standard European options, but aswe known that Black-Scholes model is the correct method of pricing theseoptions, so it is not necessary to use Monte Carlo is the formular for exact Black-Scholes model:C(s) =s2erfc( d1 2) K2e rTerfc(frac d2 2)(8)whered1=1 T[ln(sk) + (r+ 22)T],d2=d1 T(9)and erfc(x) is the complementary error function,erfc(x) =2 xe t2dt(10)Now we insert all data we have to the Black-Schole formula to check theaccuracy of our results by comparing the Monte Carlo approximation withthe value computed from exact Black-Schole formula. We generated Black-Scholes Model with parameterr= , = ,K= $50,k= 1,..,M(whereT=N t),N= 200. And we got:C(40) = ,C(45) = C(50) = The error of the Monte Carlo Estimate seems to be very we repeat the previous procedure and increased our sample size (M)to 50,000 and 100,000.

8 Then, I made a chart to check if the accuracy ofMonte Carlo simulation increases by the increasing of the sample 5:S254= , = of the accuracy of the Monte Carlo Estimate to the Black-Schole ModelMCE(1,000)MCE (10,000)MCE (50,000)MCE (100,000)Black-Schole Model C(40) = C(40) = C(40) = C(40) = (40) = C(45) = C(45) = C(45) = C(45) = (45) = C(50) = C(50) = C(50) = C(50) = (50) = the comparing Monte Carlo Estimate and the Black-Schole modle, wecan tell quite easily that the error gets smaller as the sample size we calculated the relative error by the equationMonteCarloEstimate Black ScholemodleBlack ScholeModle(11)The results areerror(1,000)error (10,000)error(50,000)error (100,000)Black-Schole ModelE(40) = (40) = (40) = (40) = (40) = (45) = (45) = (45) = (45) = (45) = (50) = (50) = (50) = (50) = (50) = : Monte Carlo simulation gives the option price is a sampleaverage, thus according to the most elementary principle of statistics, itsstandard deviation is the standard deviation of the sample divided by thesquare root of the sample size.

9 So, the error reduces at the rate of 1 overthe square root of the sample size. To sum up, the accuracy of Monte CarloSimulation is increasing by increasing the size of the 6: replaced previous value of sigma with Comparing to the Variance Reduction by An-titheric VariattesVariance Reduction by Antithetic Variatesis a simple and morewidely used way to increase the accuracy of the Monte Carlo simulation . Itis the technique used in some certain situations with an additional increasein computational complexity is the method of antithetic variates. In orderto achieve greater accuracy, one method of doing so is simple and automati-cally doubles the sample size with only a minimum increase in computationaltime. This is called theantithetic variatemethod. Because we are generat-ing obervations of a standard normal random variable which is distributedwith a mean of zero, a variance of and symmetric, there is an equallylikely chance of having drawn the observed value times 1.

10 Thus, for eacharbitrary we draw, there should be an artificially observed companion ob-servation of that can be legitimately created by us. This is the each k=1,..,M use the sequence{ (k)1,.., (k)N 1}(12)in equation (5) to simulate a payofff(Sk+1N)and also use the sequence{ (k)1,.., (k)N 1}in equation (5) to simulate an associated payofff(Sk N). Now the payoffsare simulated inpairs{f(Sk+N),f(Sk N)}This is the Mathematica Programthat we ran to evaluate the Variance comparing Monte Carlo simulation with Variance Reduction byAntithetic Variates, We made a table of 7:S254= , = (1,000)k(5,000)k(50,000)k (100,000)BS ModelV(40) = (40) = (40) = (40) = (40) = (45) = (45) = (45) = (45) = (45) = (50) = (50) = (50) = (50) = (50) = the table for the data of Monte Carlo simulation we get under thesame condition is:MCE(1,000)MCE (5,000)MCE (50,000)MCE (100,000) C(40) = C(40) = C(40) = C(40) = C(45) = C(45) = C(45) = C(45) = C(50) = C(50) = C(50) = C(50) = the chart of Monte Carlo Estimate and the Variance Reduction byAntithetic Variates, we can tell quite easily that the error gets smaller as thesample size increases.